Nuprl Lemma : member_filter_2
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. ∀x:T. ((x ∈ filter(P;L)) ⇐⇒ (x ∈ L) ∧ (↑(P x)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
true: True,
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
bool_wf,
iff_wf,
filter_wf5,
assert_wf,
list_wf,
filter_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
assert_witness,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
equal_wf,
subtype_rel_self,
set_wf,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
or_wf,
assert_elim,
not_assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
cumulativity,
because_Cache,
hypothesis,
setElimination,
rename,
functionExtensionality,
applyEquality,
productEquality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
inrFormation,
comment,
inlFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
addLevel,
impliesFunctionality,
andLevelFunctionality,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. \mforall{}x:T. ((x \mmember{} filter(P;L)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} (\muparrow{}(P x)))
Date html generated:
2017_04_14-AM-08_53_08
Last ObjectModification:
2017_02_27-PM-03_38_37
Theory : list_0
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